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Let G be a finite group. For a positive integer d, let s d N ( G ) denote the smallest integer l such that every sequence S over G of length | S | ≥ l has a nonempty product-one subsequence T with | T | ≡ 0 (mod d). In this paper, we mainly study this invariant for dihedral groups D 2 n and metacyclic groups C p ⋉ s C q .

We construct a canonical basis for the space of weakly holomorphic Drinfeld modular forms. And we find that the basis elements satisfy a generating function and the duality among coefficients of the basis elements. Moreover we obtain the congruence properties of t-expansion coefficients of these functions under some conditions.

We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup Γ H ( N , t ) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ 1 ( m ) ⋂ Γ 0 ( m N ) . Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [ Γ : Γ H ( N , t ) ] = 2 . Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.

In this paper, we investigate the joint functional distribution of Selberg zeta functions for principal congruence subgroups. We prove that the joint universality theorem for these zeta functions holds in the strip 0.85 σ 1 . As a corollary, we obtain the functional independence for the zeta functions.

We prove a congruence modulo 16 relating the class numbers h ( − 4 p ) , h ( 16 p ) of quadratic orders and Zagier's sum m ( 4 p ) associated to 4 p , when p ≡ 1 (mod 4) is a prime. This gives an analogy to Chua-Gunby-Park-Yuan's congruence established when p ≡ 3 (mod 4), and generalizes a recent work by Cheng and Guo. In particular, when p ≡ 1 (mod 4) is a prime, it is shown that the class number h ( − 4 p ) is divisible by 16 if and only if the Zagier sum m ( 4 p ) is divisible by 16.

A folklore conjecture on the partition function asserts that the density of odd values of p ( n ) is 1 2 . In general, for a positive integer t, let p t ( n ) be the t-multipartition function and δ t be the density of the odd values of p t ( n ) . It is widely believed that δ t exists. Given an odd integer a and an integer b depending on a and t, Judge and Zanello framed an infinite family of conjectural congruence relations on p t ( a n + b ) ( mod 2 ) which establishes a striking connection between δ a and δ 1 . As a special case t = 1 , it implies that δ 1 > 0 if ( 3 , a ) = 1 and δ a > 0 . This conjecture was proved for several values of a by Judge, Keith and Zanello. In this paper we prove that the conjecture is true for a = l α is a prime power with l ≥ 5 and a = 3 .

We consider the canonical basis elements f k , m e for the space of weakly holomorphic modular forms of weight k for the Hecke congruence group Γ 0 ( 2 ) and we prove that for all m ≥ c ( k ) for some constant c ( k ) , if z 0 in a fundamental domain for Γ 0 ( 2 ) is a zero of f k , m e , then either z 0 is in { i 2 , − 1 2 + i 2 , 1 2 + i 2 , − 1 + i 7 4 , 1 + i 7 4 } , or z 0 is transcendental.

Seminal works by Birch and Ihara gave formulas for the mth power moments of the traces of Frobenius endomorphisms of elliptic curves over F p for primes p ≥ 5 . Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a finite abelian group A. We revisit these formulas and determine a simple expression for the zeta function Z p ( A ; t ) , the generating function for these mth power moments. In particular, we find that Z p ( A ; t ) = Z ˆ p ( A ; t ) ∏ a ∈ Frob p ( A ) ( 1 − a t ) , where Frob p ( A ) : = { a : − 2 p ≤ a ≤ 2 p and a ≡ p + 1 ( mod | A | ) } , and Z ˆ p ( A ; t ) is an easily computed polynomial that is determined by the first ⌈ 2 ⌊ 2 p ⌋ | A | ⌉ power moments. These rational zeta functions have two natural applications. We find rational generating functions in weight aspect for traces of Hecke operators on S k ( Γ ) for various congruence subgroups Γ. We also prove congruence relations for power moments by making use of known congruences for traces of Hecke operators.

We study the question of when a congruence between two theta lifts on GSp 4 / Q descends to a congruence on modular forms on GL 2 over a quadratic field. In order to accomplish that, we use the theory of the local theta correspondence between similitude orthogonal groups and the similitude symplectic group GSp 4 , together with a classification for the degeneration modulo a prime of conductors for the L-parameters of irreducible admissible representations of GSp 4 over a non-archimedean local field. We explain that this is unlikely to be used in conjunction with existing results on congruences for GSp 4 / Q to deduce a theory of congruences over imaginary quadratic fields. On the other hand, we prove a result which does give some such congruence results by twisting.

The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly constructed density function, for the case of automorphic L-functions attached to primitive forms with respect to congruence subgroups Gamma_0(N)., Comment: 21pages

Let H l ( N ) be the set of normalized primitive holomorphic cusp forms of even integral weight l for the congruence group Γ 0 ( N ) . For any f ∈ H l ( N ) , we study the higher power moments of S f ( x ; N ) : = ∑ n ≤ x λ f ( n ) and derive the asymptotic formulas for ∫ 1 T S f k ( x ; N ) d x , k = 2 , 3 , … 7 , by using Ivic's large value arguments, where λ f ( n ) is the n-th Hecke eigenvalue of f.

Given a polynomial Q ( x 1 , ⋯ , x t ) = λ 1 x 1 k 1 + ⋯ + λ t x t k t , for every c ∈ Z and n ≥ 2 , we study the number of solutions N J ( Q ; c , n ) of the congruence equation Q ( x 1 , ⋯ , x t ) ≡ c mod n in ( Z / n Z ) t such that x i ∈ ( Z / n Z ) × for i ∈ J ⊆ I = { 1 , ⋯ , t } . We deduce formulas and an algorithm to study N J ( Q ; c , p a ) for p any prime number and a ≥ 1 any integer. As consequences of our main results, we completely solve: the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i for any prime p and any subset J of I; the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i 2 in the case t = 2 for any p and J, and the case t general for any p and J satisfying min ⁡ { v p ( λ i ) | i ∈ I } = min ⁡ { v p ( λ i ) | i ∈ J } ; the counting problem of Q ( x i ) = ∑ i ∈ I λ i x i k in the case t = 2 for any p ∤ k and any J, and in the case t general for any p ∤ k and J satisfying min ⁡ { v p ( λ i ) | i ∈ I } = min ⁡ { v p ( λ i ) | i ∈ J } .

Let n and k be nonnegative integers. In 2010, Hou and Ly evaluated the first and second moments of the n-th reversed Dickson polynomial of the first kind. In 2016, Hong, Qin and Zhao presented a recursive formula for the first moment of the n-th reversed Dickson polynomial of the second kind. In this paper, we introduce a new method to investigate the moments of the n-th reversed Dickson polynomial of ( k + 1 ) -th kind. In fact, we first show an extension of the famous Lucas' congruence and then study arithmetic properties of some two-variable linear congruences. Finally, with more efforts, we arrive at the explicit formulas for the first and second moments of the n-th reversed Dickson polynomial of the ( k + 1 ) -th kind.

We are concerned with two types of partitions considered by Andrews, Lewis and Lovejoy. One is the partitions with designated summands where exactly one is tagged among parts with equal size. The other is the partitions with designated summands where all parts are odd. In this paper, we study two partition functions P D t ( n ) and P D O t ( n ) , which count the number of tagged parts over the above two types of partitions respectively. We first give the generating functions of P D t ( n ) and P D O t ( n ) . Then we establish many congruences modulo small powers of 3 for them. Finally, we pose some problems for future work.

In this paper, we establish a new general framework under which all generalized Fibonacci-type sequences come together. The main result lies in obtaining one new general closed formula solution for all such sequences by using only matrix theory. This new formula in turn gives new closed formula solutions for most well-known sequences of this type. Next, the advantages of this formula are exploited. In particular, two applications are presented; one deals with finding the most general congruence formula for such sequences which generalizes most known formulas of this type and the other is concerned with computational number theory.

For q prime we prove a new bound for the error term in the asymptotic formula for the congruence x 1 x 2 ≡ x 3 x 4 mod q and as an application give a new bound for the 4-th moment of character sums. Our argument is based on and improves results of Ayyad, Cochrane and Zheng and Garaev and Garcia.

We study a weighted divisor function ∑ ′ m n ≤ x cos ⁡ ( 2 π m θ 1 ) sin ⁡ ( 2 π n θ 2 ) , where θ i ( 0 θ i 1 ) is a rational number. By connecting it with the divisor problem with congruence conditions, we establish an upper bound, mean-value, mean-square and some power-moments.

In the present article we extend several arithmetical results to a family of generalized Bell numbers called r-Bell numbers. In particular, we generalize some congruences such as Touchard's congruence, Sun–Zagier congruence for Bell and derangement numbers and polynomials, among others. We also describe the behavior of the 2-adic valuation of the r-Bell numbers.

In this paper, we establish a Rodriguez–Villegas type congruence for truncated q-hypergeometric functions. Using this result, we can confirm several conjectures of Guo and Zeng, such as ∑ k = 0 p − 1 ( q ; q 3 ) k ( q 2 ; q 3 ) k ( q 3 ; q 3 ) k 2 ≡ ( − 3 p ) q 1 − p 2 3 ( mod ( 1 + q + ⋯ + q p − 1 ) 2 ) , where p ⩾ 5 is a prime, ( a ; q ) n = ( 1 − a ) ( 1 − a q ) ⋯ ( 1 − a q n − 1 ) , and ( ⋅ p ) denotes the Legendre symbol modulo p.

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of $a_i=t_i=1$ $(1\leq i \leq k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit formula for the number of such solutions when $(a_1,\ldots,a_k)=t_i=1$ $(1\leq i \leq k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions., Journal of Number Theory, to appear

Let B s , t ( n ) denote the number of ( s , t ) -regular bipartitions. Recently, Dou discovered an infinite family of congruences modulo 11 for B 3 , 11 ( n ) . She also presented several conjectures on B s , t ( n ) . In this paper, utilizing an theta function identity appeared in Berndt's book, we confirm three conjectures on B 3 , 7 ( n ) given by Dou. Moreover, we prove several infinite families of congruences modulo 3 and 5 for B 3 , s ( n ) and B 5 , s ( n ) . In addition, we prove many infinite families of congruences modulo 7 for B 3 , 7 ( n ) by employing an identity of Newman.

Some congruences on conjectures of van Hamme are established. These results confirm some conjectures of Swisher.

We examine congruence relations satisfied by the fundamental unit of a pure cubic field with a power integral basis and relate those to its class number. Our approach also yields in an elementary way the congruence relations for the fundamental unit of a real quadratic field of odd class number obtained by Z. Zhang and Q. Yue in 2014.

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$. Moreover, we will present some new congruences modulo $7$ for $p_4(n)$., Comment: Final version

In this paper, we study two families of normal subgroups of Γ 0 ( 2 ) with abelian quotients and their associated modular curves. They are similar to Fermat groups and Fermat curves in some aspects but very different in other aspects. Almost all of them are non-congruence subgroups. These modular curves are either projective lines or hyperelliptic curves. There are few modular forms of weight 1 for these groups. We also determine their cuspidal divisor class groups and show that these groups are finite.

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational j-invariant. Using this, we determine the density of such real and imaginary quadratic fields. If R ( X ) denotes the number of real quadratic fields K = Q [ m ] such that | Δ K | X and for which there exists an elliptic curve E / K with rational j-invariant that has everywhere good reduction, then R ( X ) ≫ X log ⁡ ( X ) . We also obtain a similar result for imaginary quadratic fields. To obtain these estimates we explicitly construct quadratic fields over which we can construct elliptic curves with everywhere good reduction. The estimates then follow from elementary multiplicative number theory. In addition, we obtain infinite families of real and imaginary quadratic fields such that there are no elliptic curves with everywhere good reduction over these fields.

Let p be an odd prime. We show that the integral points on the sphere with radius n are equidistributed modulo p as n ⟶ ∞ where n is not of the shape 4 l ( 8 m + 7 ) and its 2-adic valuation is bounded. In particular if n is sufficiently large and if n satisfies a congruence equation α 1 2 + α 2 2 + α 3 2 ≡ n ( mod p ) where p 2 | n if all α i ≡ 0 ( mod p ) , then there are integers x i with x i ≡ α i ( mod p ) ( i = 1 , 2 , 3 ) satisfying x 1 2 + x 2 2 + x 3 2 = n . The similar result holds also in the case modulo 8.

We prove that in most cases the Jacquet-Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.

The numbers $R_n$ and $W_n$ are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\ W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We prove that, for any positive integer $n$ and odd prime $p$, there hold \begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\ 9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2 &\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if $p>3$.} \end{align*} The first two congruences were originally conjectured by Z.-W. Sun. Our proof is based on the multi-variable Zeilberger algorithm and the following observation: $$ {2n\choose n}{n\choose k}{m\choose k}{k\choose m-n}\equiv 0\pmod{{2k\choose k}{2m-2k\choose m-k}}, $$ where $0\leqslant k\leqslant n\leqslant m \leqslant 2n$., 18 pages

Let P n denote the set of positive integers which are prime to n. Let B n be the n-th Bernoulli number. For any prime p > 5 and integer r ≥ 2 , we prove that ∑ l 1 + l 2 + ⋯ + l 5 = p r l 1 , ⋯ , l 5 ∈ P p 1 l 1 l 2 l 3 l 4 l 5 ≡ − 5 ! 6 p r − 1 B p − 5 ( mod p r ) . This gives an extension of a family of curious congruences found by the author, Cai and Zhao.

We prove that if a prime l > 3 divides p k − 1 , where p is prime, then there is a congruence modulo l, like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate l to primes like 691 by viewing it as a divisor of a partial zeta value, and see how a construction of Ribet links the congruence with the Bloch–Kato conjecture (theorem in this case). This viewpoint allows us to give a new proof of a recent theorem of Billerey and Menares. We end with some examples, including where p = 2 and l is a Mersenne prime.

Let Δ k ( n ) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Radu and Sellers conjectured that for all α ⩾ 1 and n ⩾ 0 , Δ 3 ( λ α ) Δ 3 ( 2 α + 2 n + λ α + 2 ) ≡ Δ 3 ( λ α + 2 ) Δ 3 ( 2 α n + λ α ) ( mod 2 α ) , where λ α = 2 α + 1 + 1 3 if α is even and λ α = 2 α + 1 3 if α is odd. Radu and Sellers proved that this conjecture is true for α = 1 . In this work, we show that this conjecture holds for α = 2 . We also prove that Δ 3 ( λ α ) ≡ ( − 1 ) [ α 2 ] ( mod 4 ) which yields Δ 3 ( λ α ) ≡ 1 ( mod 2 ) . This congruence was conjectured by Radu and Sellers. Furthermore, we also deduce some new Ramanujan-type congruences modulo 2 and 4 for 7-core partitions.

This paper proves a result concerning linear forms on the Sinnott module. This is perhaps of intrinsic interest, and it is needed in another paper of the same authors. We obtain a congruence which can be interpreted as a strengthening of a congruence of Anthony Hayward.

In this paper, we investigate the arithmetic properties of bipartitions with 3-core. Let A 3 ( n ) denote the number of bipartitions with 3-core of n. We will prove one infinite family of congruences modulo 5 for A 3 ( n ) . We also establish one surprising congruence modulo 14 for A 3 ( 8 n + 6 ) . Finally, we prove that, if u ( n ) denotes the number of representations of a nonnegative integer n in the form x 2 + y 2 + 3 z 2 + 3 t 2 with x , y , z , t ∈ Z , then u ( 6 n + 5 ) = 12 A 3 ( 2 n + 1 ) .

We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A of C , and specifically the highest weight needed to generate these rings as A-algebras. In particular, we determine upper bounds, independent of N, for the highest needed weight that generates the C -algebras of modular forms over Γ 1 ( N ) and Γ 0 ( N ) with some conditions on N. For N ⩾ 5 , we prove that the Z [ 1 / N ] -algebra of modular forms over Γ 1 ( N ) with coefficients in Z [ 1 / N ] is generated in weight at most 3. We give an algorithm that computes the generators, and supply some computations that allow us to state two conjectures concerning the situation over Γ 0 ( N ) .

Let K = Q ( d ) be a real quadratic field with odd class number and its fundamental unit ϵ d = x + y d > 1 satisfies N K / Q ( ϵ d ) = 1 . We give some congruence relations about x , y explicitly.

The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n∈Z+ of Pascalʼs triangle is captured by the so-called Gaussian formula, which states that ∑m=0n(−1)m(nm)q is 0 if n is odd, and is equal to ∏kodd(1−qk) if n is even. In this paper, we find a q-binomial congruence which synthesizes this result and Fleckʼs congruence for binomial coefficients, which asserts that for n,p∈Z+, with p a prime, ∑m≡j(modp)(−1)m(nm)≡0(modp⌊n−1p−1⌋).

The coefficient series of the holomorphic Picard–Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard–Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their associated Picard–Fuchs solutions lie in Z 〚 t 〛 . A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin–Swinnerton-Dyer congruence is proven for the coefficient series attached to Γ 1 ( 7 ) . We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form u n ∼ l λ n / n . Certain arithmetic results extend to the study of general holonomic recurrences.

Text We prove two congruence results concerning Kloosterman sums over finite fields. The first result concerns the coefficients of the characteristic polynomial over Q of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27. We use methods from algebraic number theory such as Stickelbergerʼs theorem and the Gross–Koblitz formula, as well as Fourier analysis. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VJcB6W_PQ0s .

For every integer k, a k-representation of 2 k − 1 is a string n = ( n 1 , … , n k ) of nonnegative integers such that ∑ j = 1 k 2 n j = 2 k − 1 , and W ( 1 , k ) is their number. We present an efficient recursive formula for W ( 1 , k ) ; this formula allows also to prove the congruence W ( 1 , k ) = 4 + ( − 1 ) k ( mod 8 ) for k ⩾ 3 .

Several authors have studied the nature and location of zeros of modular forms for the full modular group Γ and other congruence subgroups. In this paper, we investigate the zeros of certain quasi-modular forms for Γ . In particular, we study the transcendence and existence of infinitely many Γ -inequivalent zeros of these quasi-modular forms. We also estimate the number of such zeros in Siegel sets, motivated by a recent work of Ghosh and Sarnak.

We determine the zeroes of Drinfeld–Goss Eisenstein series for the principal congruence subgroups Γ ( N ) of Γ = GL ( 2 , F q [ T ] ) on the Drinfeld modular curve X ( N ) .

We give a general arithmetic dimension formula for spaces of vector-valued Siegel cusp forms of degree two. Then, using this formula, we derive explicit dimension formulas for arithmetic subgroups of any level for each Q -form of Sp ( 2 ; R ) . Tsushima has already given the dimension formulas for some congruence subgroups of the split Q -form in Tsushima (1983, 1997) [32] , [33] . We obtain an alternative proof for his results by using the Selberg trace formula and the theory of prehomogeneous vector spaces. As for the non-split Q -forms, our results are new. We generalize the results and proofs given in Arakawa (1981) [1] , Christian (1969, 1975, 1977) [5] , [6] , Hashimoto (1983, 1984) [12] , [13] , Morita (1974) [25] for the scalar-valued case to the vector-valued case using the Selberg trace formula.

We estimate the deviation of the number of solutions of the congruence m 2 − n 2 ≡ c ( mod q ) , 1 ⩽ m ⩽ M , 1 ⩽ n ⩽ N , from its expected value on average over c = 1 , … , q . This estimate is motivated by the connection, recently established by D.R. Heath-Brown, between the distribution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function α k 2 , k = 1 , 2 , … with a real α .

Let rs(n) denote the number of representations of n as the sum of s squares of integers. In this note, we determine r2ks(n) modulo 2k+2 for k⩾1 and s odd. For general s, we also obtain a congruence for rs(n) modulo 2s. These extend Wagstaff's results (J. Number Theory 127 (2007) 326–329).

The ‘Congruence Conjecture’ was developed by the second author in a previous paper [So3]. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields.

For any positive integer m we obtain the asymptotic formula, | B ∩ V ′ | = | B ′ | ϕ ( m ) + O ( 8 ν ( m ) τ ( m ) ( log m ) 3 ( log log m ) 7 | B | ) , for the number of solutions of the congruence x 1 x 2 ≡ x 3 x 4 ( mod m ) with coordinates relatively prime to m in a box B of arbitrary size and position. We also obtain an upper bound for a fourth-order character sum moment, 1 ϕ ( m ) ∑ χ ≠ χ 0 | ∑ x = a + 1 a + B χ ( x ) | 4 ≪ 8 ν ( m ) τ ( m ) ( log m ) 3 ( log log m ) 7 B 2 .

Let K be an abelian extension of a totally real number field k, K + its maximal real subfield and G = Gal ( K / k ) . We have previously used twisted zeta-functions to define a meromorphic C G -valued function Φ K / k ( s ) in a way similar to the use of partial zeta-functions to define the better-known function Θ K / k ( s ) . For each prime number p, we now show how the value Φ K / k ( 0 ) combines with a p-adic regulator of semilocal units to define a natural Z p G -submodule of Q p G which we denote S K / k . If p is odd and splits in k, our main theorem states that S K / k is (at least) contained in Z p G . Thanks to a precise relation between Φ K / k ( 1 − s ) and Θ K / k ( s ) , this theorem can be reformulated in terms of (the minus part of) Θ K / k ( s ) at s = 1 , making it an analogue of Deligne–Ribet and Cassou-Nogues' well-known integrality result concerning Θ K / k ( 0 ) . We also formulate some conjectures including a congruence involving Hilbert symbols that links S K / k with the Rubin–Stark Conjecture for K + / k .

We determine rs(n) modulo 2s when s is a prime or a power of 2. For general s, we prove a congruence for rs(n) modulo the largest power of 2 dividing 2s.

Matrix representations of Hecke operators on classical holomorphical cusp forms and the corresponding period polynomials are well known. In this article we derive representations of Hecke operators for vector valued period functions for the congruence subgroups 0 (n). For this we use an integral transform from the space of vector valued cusp forms to the space of vector valued period functions.